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Let $x,y\in X$.

Does strictly convex preference (which implies that the utility is strictly quasiconcave) mean that:

$x\succsim y$ implies $\alpha x+(1-\alpha)y\succ y$ for any $\alpha\in (0,1)$?

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    $\begingroup$ yes, if you add the assumption that $x \ne y$. $\endgroup$ – tdm May 2 at 9:33

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